Mazur’s structure theorem
Any normed associative real division algebra is isomorphic^{} to one of the following:

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The real numbers $\mathbb{R}$.

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The complex numbers^{} $\u2102$.

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The quaternions $\mathbb{H}$.
The next generalization^{}, the octonions, often viewed as the “complexification” of the quaternions, fails to be associative.
Title  Mazur’s structure^{} theorem 

Canonical name  MazursStructureTheorem 
Date of creation  20130322 14:50:04 
Last modified on  20130322 14:50:04 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Theorem 
Classification  msc 11R52 
Related topic  TheoremsOnSumsOfSquares 